Vitali convergence theorem
Let ${f}_{1},{f}_{2},\mathrm{\dots}$ be ${\mathbf{L}}^{p}$integrable functions on some measure space^{}, for $$.
The sequence $\{{f}_{n}\}$ converges^{} in ${\mathbf{L}}^{p}$ to a measurable function^{} $f$ if and and only if

i
the sequence $\{{f}_{n}\}$ converges to $f$ in measure;

ii
the functions $\{{{f}_{n}}^{p}\}$ are uniformly integrable; and

iii
for every $\u03f5>0$, there exists a set $E$ of finite measure, such that $$ for all $n$.
Remarks
This theorem can be used as a replacement for the more wellknown dominated convergence theorem, when a dominating cannot be found for the functions ${f}_{n}$ to be integrated. (If this theorem is known, the dominated convergence theorem can be derived as a special case.)
In a finite measure space, condition (iii) is trivial. In fact, condition (iii) is the tool used to reduce considerations in the general case to the case of a finite measure space.
In probability , the definition of “uniform integrability” is slightly different from its definition in general measure theory; either definition may be used in the statement of this theorem.
References
 1 Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications, second ed. WileyInterscience, 1999.
 2 Jeffrey S. Rosenthal. A First Look at Rigorous Probability Theory. World Scientific, 2003.
Title  Vitali convergence theorem^{} 

Canonical name  VitaliConvergenceTheorem 
Date of creation  20130322 16:17:10 
Last modified on  20130322 16:17:10 
Owner  stevecheng (10074) 
Last modified by  stevecheng (10074) 
Numerical id  9 
Author  stevecheng (10074) 
Entry type  Theorem 
Classification  msc 28A20 
Synonym  uniformintegrability convergence theorem 
Related topic  ModesOfConvergenceOfSequencesOfMeasurableFunctions 
Related topic  UniformlyIntegrable 
Related topic  DominatedConvergenceTheorem 